User:Unque/Barbershop Tuning Theory

While Barbershop music is typically said to use 7-limit Just Intonation, in practice it contains a number of idiosyncrasies and dialectal variation that differ from one performer to another, often dipping into tempered tunings or even into higher prime limits. On this page, I plan to document the chords of Barbershop harmony and the variations that occur among them; additionally, I will discuss the theoretical reasoning behind a performer's choice of one tuning over another, and theoretical extensions that could provide other potential tunings for a given chord. Any purely theoretical or otherwise unattested tunings will be explicitly disclaimed as such.

Note that the ratios given here are approximations, not precise measurements. Most performers don't analyze the intervals as JI ratios (with the common exception of certain extremely simple chords, such as the famous 4:5:6:7 tetrad), but rather slightly adjust the tunings to make the chords ring better. Even if they did analyze the ratios, the voice would not be precise enough to articulate certain minute differences, and as such these ratios would still not precisely represent the actual practical sounds being sung.

Chords are given here as harmonic segments within the octave unless otherwise specified; true Barbershop performance often uses more open voicings, inversions, or other kinds of permutations with respect to the equivalent octaves.

The Barbershop Seventh

The Major Triad is listed by the Barbershop Harmony Society as the most consonant chord in the Barbershop vocabulary, and the Barbershop Seventh is considered its canonical extension into a tetrad.

The 4:5:6:7 tetrad is almost entirely consistent among all performers; however, there are some uncommon instances of the Barbershop Seventh being tuned sharp to better imitate a Dominant function.

Barbershop Seventh Tunings

Segment	Ratios	Step Sizes	Comments
4:5:6:7	5/4, 3/2, 7/4	5/4, 6/5, 7/6, (8/7)	Widely agreed to be the most consonant tuning. Extremely common, and characterizes Barbershop music.
20:25:30:36	5/4, 3/2, 9/5	5/4, 6/5, 6/5, (10/9)	Less consonant, but clearer dominant function. Particularly rare.

See also: Septimal Meantone, a version of Meantone that finds 7/4 at the Augmented Sixth.

Half-Diminished Seventh (and its inversions)

The Half-Diminished Seventh tetrad is officially considered by the Barbershop Harmony Society to be the second-most consonant chord (after the Barbershop Seventh, of course); in common practice 12-EDO, however, this chord is far from consonant, especially compared to other chords that may compete for that title. This difference in interpretation must be owed to conventions either in compositional use or in tuning - so, how do vocal performers tend to sing Half-Diminished Seventh chords?

The tuning of the Half-Diminished Seventh chord and its inversions are some of the most controversial in Barbershop music, especially due to differing preferences for "rooting" the chord on a specific interval. In general, chords tend to feel most rooted when their harmonic segment forms are simplest; for instance, the 8 in 6:7:8:10 sounds more like the root than the 6 does, because the simplest inversion is 4:5:6:7.

If one wishes to root the chord as a Half-Diminished Seventh, then inversions of 5:6:7:9 are the most common interpretation. If one instead wishes to root the chord as a Minor triad with an added Major Sixth, the chord is most commonly tuned as a standard 1/(6:5:4) Minor Triad with some type of Major Sixth (typically 5/3, or less commonly 12/7) added on top.

It should additionally be noted that the ii^ø7 chord is used as a dominant due to being the negative harmony variant of V⁷. Because of this, some performers prefer to treat this chord as an undertonal variant of the standard Barbershop Seventh.

Finally, the half-diminished chord can be seen as a minor tetrad with a lowered fifth degree; this can yield 5:6:7:9 if we consider the 6/5 minor third, or 12:14:17:21 if we consider the 7/6 minor third.

Half-Diminished Tunings

Segment	Ratios	Step Sizes	Comments
5:6:7:9	6/5, 7/5, 9/5	6/5, 7/6, 9/7, (10/9)	The most consonant tuning of this chord.
25:30:36:45	6/5, 36/25, 9/5	6/5, 6/5, 5/4, (10/9)	1/(6:5:4) triad stacked on top of a 5:6 dyad. Undertonal inversion of Dominant chord.
60:70:84:105	7/6, 7/5, 7/4	7/6, 6/5, 5/4, (8/7)	1/(6:5:4) triad stacked on top of a 6:7 dyad. Undertonal inversion of Barbershop 7th chord.
17:20:24:30	20/17, 24/17, 30/17	20/17, 6/5, 5/4, (17/15)	1/(6:5:4) triad stacked on top of a 17:20 dyad. Most likely conceptualized as tempered tuning, not 17-limit JI.
12:14:17:21	7/6, 17/12, 7/4	7/6, 17/14, 21/17, (8/7)	12:14:18:21 tetrad, with the fifth lowered. Most likely conceptualized as tempered tuning, not 17-limit JI.

See also: 5afdo and 7ifdo, the low-complexity JI scales from which several of these tunings are derived.

Add-Nine Chords

The Major Add9 tetrad (and by extension, the sus2 triad) is officially considered as part of the Barbershop vocabulary, but is not treated as fundamental on the same level as the previous two chords. This chord is much less variable than some of the other chords, with the ninth being almost unanimously considered a 9/4 above the tonic.

Major Ninth Tunings

Segment	Ratios	Step Sizes	Comments
4:5:6:9	5/4, 3/2, 9/4	5/4, 6/5, 3/2, (16/9)	Open voicing with ninth on top.
8:9:10:12	9/8, 5/4, 3/2	9/8, 10/9, 6/5, (4/3)	Close voicing with ninth reduced.

Minor Chords

The Minor triad and tetrad are relatively agreed-upon in its tuning methods. The form 1/(6:5:4) or 10:12:15 is used most often, and is considered by most to be the canonical tuning of the Minor triad; however, many performers criticize the 6/5 for its sharpness, and as such prefer to use tempered versions that more closely resemble the 1\4 Minor Third found more often in contemporary music. Some performers use a septimal minor third to preserve the "ringing" effect of Just Intonation, yielding a 6:7:9 chord, but this is not particularly common.

For the tetrad, the form 10:12:15:18 is used as an extension of the 1/(6:5:4) triad, but the seventh is considered somewhat discordant in this tuning. Extending the 7-limit triad instead yields 12:14:18:21, which has a more consonant seventh and a flatter third, but is significantly more complex than the 5-limit form. Of the two, the 5-limit form is significantly more popular due to being easier to tune by ear and generally more recognizable in quality.

Minor Seventh Tunings

Segment	Ratios	Step Sizes	Comments
10:12:15:18	6/5, 3/2, 9/5	6/5, 5/4, 6/5, (10/9)	Standard 5-limit Minor Tetrad; much more common than 7-limit.
12:14:18:21	7/6, 3/2, 7/4	7/6, 9/7, 7/6, (8/7)	Standard 7-limit Minor Tetrad; much less common than 5-limit.
32:38:48:57	19/16, 3/2, 57/32	19/16, 24/19, 19/16, (64/57)	Much closer to 12edo's minor third. Most likely conceptualized as tempered tuning, not 19-limit JI.

Major Seventh

Although officially considered as part of the Barbershop vocabulary, the Major Seventh tetrad is officially recognized as a dissonance due to the "grinding" effect that occurs between the root and the seventh.

The seventh is almost always tuned as 15/8 above the root, though it is occasionally tempered to reduce grinding.

Major Seventh Tunings

Segment	Ratios	Step Sizes	Comments
8:10:12:15	5/4, 3/2, 15/8	5/4, 6/5, 5/4, (16/15)	This is the only interpretation of the chord that I could find attestation of in Barbershop music.

Full Diminished Seventh

The Full Diminished tetrad is rarely used in Barbershop music, but is considered a part of the vocabulary by the Barbershop Harmony Society nonetheless. The simplest way to tune this tetrad is by stacking up intervals of 6/5, but this becomes complex and dissonant quite quickly.

Another option is to use the Half-Diminished tetrad, but flatten the seventh to yield a Full Diminished form; in order to do this, however, one must have a determined tuning for the Half-Diminished Seventh chord, which is already controversial. Additionally, the amount by which the seventh is flattened varies idiolectally, since there is no low-complexity tuning for the Diminished Seventh interval aside from the enharmonic Major Sixth.

Due to the Diminished Chord's nature as a symmetrical chord in Western classical practices, rotations of the tetrad are also valid options.

Full Diminished Tunings

Segment	Ratios	Step Sizes	Comments
125:150:180:216	6/5, 36/25, 216/125	6/5, 6/5, 6/5, (125/108)	Made by stacking 6/5 continually. Extremely complex and dissonant, but easy to tune.
15:18:21:25	6/5, 7/5, 5/3	6/5, 7/6, 25/21, (6/5)	Diminished triad with added Major Sixth.
15:18:21:26	6/5, 7/5, 26/15	6/5, 7/6, 26/21, (15/13)	Most likely conceptualized as tempered tuning, not 13-limit JI.
10:12:14:17	6/5, 7/5, 17/10	6/5, 7/6, 17/14, (20/17)	Most likely conceptualized as tempered tuning, not 17-limit JI.

See also: Kleismic and Diminished, temperaments who provide essentially-tempered versions of the diminished tetrad.

Dominant Diminished

While the Dominant Diminished tetrad (a dominant chord with a diminished fifth) is officially recognized as part of the Barbershop vocabulary, it is rarely used in practice, and as such there are very few attestations of how to tune it. It is also called a Major Diminished chord, or simply a Flat-Five chord.

The Major Diminished chord can be seen as a barbershop tetrad with the fifth diminished; this interpretation yields the tuning 20:25:28:35. It can also be thought of as a Half-Diminished tetrad with the third raised, which may instead yield 20:25:28:36.

Additionally, rooting a Half-Diminished chord on the perfect fifth rather than the root yields a chord with intervals 1 - M3 - Aug4 - M6, which is very close to a Dominant Diminished chord, especially considering how sharp the M6 is tuned and how flat the dominant seventh is in Barbershop music. The major sixth can be raised to a typical minor seventh in tunings where it would be more beneficial to do so.

It can also be thought of as a Lydian Dominant chord with the fifth omitted; thus, 8:10:11:14 or 4:5:7:11. This tuning is seldom used in Barbershop practice, but is often used as the tuning of this chord in harmonic contexts.

Dominant Diminished Tunings

Segment	Ratios	Step Sizes	Comments
20:25:28:35	5/4, 7/5, 7/4	5/4, 28/25, 5/4, (8/7)	Tuned such that the ratio between the Barbershop Seventh and the Diminished Fifth is 5/4.
20:25:28:36	5/4, 7/5, 9/5	5/4, 28/25, 9/7, (10/9)	Half-diminished chord with raised third.
7:9:10:12	9/7, 10/7, 12/7	9/7, 10/9, 6/5, (7/6)	Rotation of half-diminished chord.
12:15:17:21	5/4, 17/12, 7/4	5/4, 17/15, 21/12, (8/7)	17-limit half-diminished chord, but with the sixth raised.
10:13:14:18	13/10, 7/5, 9/5	13/10, 14/13, 9/7, (10/9)	Purely theoretical; despite ringing very well, 13/10 is too sharp to be expected of typical Barbershop music.
15:19:21:27	19/15, 7/5, 9/5	19/15, 21/19, 9/7, (10/9)	Most likely conceptualized as tempered tuning, not 19-limit JI.
4:5:7:11	5/4, 7/4, 11/4	5/4, 7/5, 11/7, (16/11)	Lydian chord with the fifth omitted. Open voicing with eleventh on top.
8:10:11:14	5/4, 11/8, 7/4	5/4, 11/10, 14/11, (8/7)	Close voicing with eleventh reduced.

See also: 8afdo, the JI scale from which Lydian Dominant is derived.

Augmented Chord

The Augmented triad is made by stacking three Major Thirds on top of each other; due to its nature as a symmetrical chord in classical theory, it may also be inverted to include a Minor Sixth in place of the fifth, or even a Diminished Fourth in place of the Major Third.

The only attested tuning of this chord seems to be permutations of 16:20:25, though it may be tempered to better approximate 3-EDO.

Alternatively, one of the thirds could theoretically be tuned a flat by a Magic Comma in order to add the more consonant 9/7 on top of the chord; while this is technically valid and rings very well, it is purely theoretical and does not seem to be attested as a tuning of the Augmented triad in barbershop.

Some performers add the Dominant Seventh (9/5) or a Barbershop Seventh (7/4) over the top of an Augmented Triad, but this is usually used as a flourish and not considered a part of the chord.

Augmented Chord Tuning

Segment	Ratios	Step Sizes	Comments
16:20:25:32	5/4, 25/16, 2/1	5/4, 5/4, 32/25	The definitive JI augmented chord.
12:15:19:24	5/4, 19/12, 2/1	5/4, 19/15, 24/19	Much closer to 3-EDO. Most likely conceptualized as tempered tuning, not 19-limit JI.
28:36:45:56	9/7, 45/28, 2/1	9/7, 5/4, 56/45	Purely theoretical. Seemingly unattested, despite ringing very well.

See also: Augmented, Dudon, and Magic, temperaments whose characteristic essentially tempered chords approximate the ones shown here.